Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-06-08T14:46:51.160Z Has data issue: false hasContentIssue false
  • English
  • Français

A Polynomial Algorithm for Constructing a Large Bipartite Subgraph, with an Application to a Satisfiability Problem

Published online by Cambridge University Press:  20 November 2018

Svatopluk Poljak  and
Daniel Turzík
Svatopluk Poljak
Affiliation:
Technical University, Prague, Czechoslovakia
Daniel Turzík
Affiliation:
Institute of Chemical Technology, Prague, Czechoslovakia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a symmetric connected graph without loops. Denote by b(G) the maximum number of edges in a bipartite subgraph of G. Determination of b(G) is polynomial for planar graphs ([6], [8]); in general it is an NP-complete problem ([5]). Edwards in [1], [2] found some estimates of b(G) which give, in particular,

for a connected graph G of n vertices and m edges, where

and ﹛x﹜ denotes the smallest integer ≧ x.

We give an 0(V3) algorithm which for a given graph constructs a bipartite subgraph B with at least f(m, n) edges, yielding a short proof of Edwards’ result.

Further, we consider similar methods for obtaining some estimates for a particular case of the satisfiability problem. Let Φ be a Boolean formula of variables x1, …, xn.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 3 , 01 June 1982 , pp. 519 - 524
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Edwards, C. S., Some extremal properties of bipartite subgraphs, Can. J. Math. 25 (1973), 475–485.Google Scholar
2. Edwards, C. S., An improved lower bound for the number of edges in a largest bipartite subgraph, in: Recent advances in graph theory (Academia, Prague), 1975–167.Google Scholar
3. S. A., Cook, The complexity of theorem-proving procedures, Proc. 3rd Ann. ACM Symp. on Theory of Computing, Association for Computing Machinery, New York (1970), 151–158.Google Scholar
4. Davis, M. and Putnam, H., A computing procedure for quantification theory, JACM 7 (1960), 201–215.Google Scholar
5. Garey, M. R., Johnson, D. S. and Stockmayer, L., Some simplified NP-complete graph problems, Theoretical Computer S ience 1 (1976), 237–267.Google Scholar
6. Hadlock, R. O., Finding a maximum cut of a planar graph in polynomial time, SIAM J. Comp. 4 (1975), 221–225.Google Scholar
7. Harary, F., Graph theory (Addison Wesley, Reading, Massachusetts, 1969.Google Scholar
8. Orlova, G. I. and Dorfman, Y. G., Finding the maximum cut in a graph, Engrg. Cybernetics 10 (1972), 502–506.Google Scholar
9. Paton, K., An algorithm for the blocks and cutnodes of a graph, Comm. ACM 14 (1971), 468–476.Google Scholar
10. Tarjan, R., Depth-first search and linear graph algorithms, SIAM J. Comput. 1 (1972), 146–160.Google Scholar
11. Tarjan, R., An efficient planarity algorithm, Thesis, Stanford University (1971).Google Scholar